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Theorem wl-ifp-ncond1 34948
 Description: If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.)
Assertion
Ref Expression
wl-ifp-ncond1 𝜓 → (if-(𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑𝜒)))

Proof of Theorem wl-ifp-ncond1
StepHypRef Expression
1 df-ifp 1059 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 simpr 488 . . . 4 ((𝜑𝜓) → 𝜓)
32con3i 157 . . 3 𝜓 → ¬ (𝜑𝜓))
4 biorf 934 . . 3 (¬ (𝜑𝜓) → ((¬ 𝜑𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒))))
53, 4syl 17 . 2 𝜓 → ((¬ 𝜑𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒))))
61, 5bitr4id 293 1 𝜓 → (if-(𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059 This theorem is referenced by:  wl-ifp-ncond2  34949
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